Suppose that 75% of all voters prefer Candidate A.
If 7 people are chosen at random for a poll, what is the probability that exactly 2 of them favor Candidate A?
7C2 * .75^2 * .25^5
To find the probability that exactly 2 out of 7 randomly chosen people favor Candidate A, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of getting exactly k successes
C(n, k) is the number of combinations of n items taken k at a time
p is the probability of success (75% or 0.75)
n is the total number of trials (7 in this case)
k is the number of successes (2 in this case)
Using these values, we can calculate the probability as follows:
P(X = 2) = C(7, 2) * (0.75)^2 * (1-0.75)^(7-2)
Now let's calculate each part of the formula step-by-step:
C(7, 2) = 7! / (2! * (7-2)!) = 7! / (2! * 5!) = (7 * 6) / (2 * 1) = 21
(0.75)^2 = 0.5625
(1-0.75)^(7-2) = 0.25^5 = 0.001953125
Now let's substitute these values into the formula:
P(X = 2) = 21 * 0.5625 * 0.001953125
P(X = 2) = 0.02181884765625
Therefore, the probability that exactly 2 out of 7 randomly chosen people favor Candidate A is approximately 0.0218 or 2.18%.
To calculate the probability of exactly 2 out of 7 randomly chosen people favoring Candidate A, you need to use the concept of binomial probability.
Binomial probability refers to the probability of exactly 𝑘 successes in 𝑛 independent Bernoulli trials, where each trial has a probability 𝑝 of success and 1−𝑝 of failure.
In this case, 𝑛 represents the total number of people chosen, which is 7, and 𝑘 represents the number of people who favor Candidate A, which is 2. The probability of success, 𝑝, is given as 0.75 (or 75%).
The formula to calculate binomial probability is as follows:
𝑃(𝑋 = 𝑘) = 𝐶(𝑛, 𝑘) × 𝑝^𝑘 × (1−𝑝)^(𝑛−𝑘)
Where:
- 𝑃(𝑋 = 𝑘) is the probability of exactly 𝑘 successes,
- 𝐶(𝑛, 𝑘) is the binomial coefficient, which represents the number of ways to choose 𝑘 items from 𝑛 items, and is calculated as 𝐶(𝑛, 𝑘) = 𝑛! / (𝑘! × (𝑛−𝑘)!),
- 𝑝^𝑘 represents the probability of 𝑘 successes,
- (1−𝑝)^(𝑛−𝑘) represents the probability of 𝑛−𝑘 failures.
Using this formula, you can plug in the values to calculate the probability:
𝑃(𝑋 = 2) = 𝐶(7, 2) × 0.75^2 × (1−0.75)^(7−2)
Calculating the binomial coefficient:
𝐶(7, 2) = 7! / (2! × (7−2)!)
= 7! / (2! × 5!)
𝐶(7, 2) = (7 × 6) / (2 × 1)
= 21
Plugging this value and the other probability values into the formula:
𝑃(𝑋 = 2) = 21 × 0.75^2 × (1−0.75)^(7−2)
𝑃(𝑋 = 2) = 21 × 0.5625 × 0.0439
𝑃(𝑋 = 2) ≈ 0.512
Therefore, the probability that exactly 2 out of 7 randomly chosen people favor Candidate A is approximately 0.512, or 51.2%.